LMFDB - Embedded newform 1300.2.c.c.1249.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1300,2,Mod(1249,1300)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1300, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1300.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1300 = 2^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1300.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$10.3805522628$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 52)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1300.1249
Dual form			1300.2.c.c.1249.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{7} +3.00000 q^{9} +O(q^{10})$	$q-2.00000i q^{7} +3.00000 q^{9} -2.00000 q^{11} +1.00000i q^{13} +6.00000i q^{17} +6.00000 q^{19} -8.00000i q^{23} -2.00000 q^{29} +10.0000 q^{31} -6.00000i q^{37} -6.00000 q^{41} -4.00000i q^{43} -2.00000i q^{47} +3.00000 q^{49} -6.00000i q^{53} +10.0000 q^{59} -2.00000 q^{61} -6.00000i q^{63} +10.0000i q^{67} +10.0000 q^{71} -2.00000i q^{73} +4.00000i q^{77} +4.00000 q^{79} +9.00000 q^{81} +6.00000i q^{83} +6.00000 q^{89} +2.00000 q^{91} +2.00000i q^{97} -6.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{9}+O(q^{10}) $ 2 * q + 6 * q^9	$ 2 q + 6 q^{9} - 4 q^{11} + 12 q^{19} - 4 q^{29} + 20 q^{31} - 12 q^{41} + 6 q^{49} + 20 q^{59} - 4 q^{61} + 20 q^{71} + 8 q^{79} + 18 q^{81} + 12 q^{89} + 4 q^{91} - 12 q^{99}+O(q^{100}) $ 2 * q + 6 * q^9 - 4 * q^11 + 12 * q^19 - 4 * q^29 + 20 * q^31 - 12 * q^41 + 6 * q^49 + 20 * q^59 - 4 * q^61 + 20 * q^71 + 8 * q^79 + 18 * q^81 + 12 * q^89 + 4 * q^91 - 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1300\mathbb{Z}\right)^\times$.

$n$	$301$	$651$	$677$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	0	0
$9$	3.00000	1.00000
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.00000i	1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$	6.00000i	1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 8.00000i	− 1.66812i	−0.551677	−	0.834058i	$-0.686012\pi$
$23$	− 8.00000i	− 1.66812i	0.551677	−	0.834058i	$-0.313988\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	10.0000	1.79605	0.898027	−	0.439941i	$-0.145001\pi$
$31$	10.0000	1.79605	0.898027	+	0.439941i	$0.145001\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 6.00000i	− 0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$	− 6.00000i	− 0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 2.00000i	− 0.291730i	−0.989305	−	0.145865i	$-0.953403\pi$
$47$	− 2.00000i	− 0.291730i	0.989305	−	0.145865i	$-0.0465965\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.0000	1.30189	0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000	1.30189	0.650945	+	0.759125i	$0.274373\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	− 6.00000i	− 0.755929i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.0000i	1.22169i	0.791748	+	0.610847i	$0.209171\pi$
$67$	10.0000i	1.22169i	−0.791748	+	0.610847i	$0.790829\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.0000	1.18678	0.593391	−	0.804914i	$-0.297789\pi$
$71$	10.0000	1.18678	0.593391	+	0.804914i	$0.297789\pi$
$72$	0	0
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.00000i	0.455842i
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	6.00000i	0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$	6.00000i	0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	0	0
$99$	−6.00000	−0.603023
$100$	0	0
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	0	0
$103$	8.00000i	0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$	8.00000i	0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 16.0000i	− 1.54678i	−0.633932	−	0.773389i	$-0.718560\pi$
$107$	− 16.0000i	− 1.54678i	0.633932	−	0.773389i	$-0.281440\pi$
$108$	0	0
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 14.0000i	− 1.31701i	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	− 14.0000i	− 1.31701i	0.752577	−	0.658505i	$-0.228811\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.00000i	0.277350i
$118$	0	0
$119$	12.0000	1.10004
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 8.00000i	− 0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$	− 8.00000i	− 0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−16.0000	−1.39793	−0.698963	−	0.715158i	$-0.746355\pi$
$131$	−16.0000	−1.39793	−0.698963	+	0.715158i	$0.746355\pi$
$132$	0	0
$133$	− 12.0000i	− 1.04053i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	18.0000i	1.53784i	0.639343	+	0.768922i	$0.279207\pi$
$137$	18.0000i	1.53784i	−0.639343	+	0.768922i	$0.720793\pi$
$138$	0	0
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 2.00000i	− 0.167248i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	6.00000	0.488273	0.244137	−	0.969741i	$-0.421495\pi$
$151$	6.00000	0.488273	0.244137	+	0.969741i	$0.421495\pi$
$152$	0	0
$153$	18.0000i	1.45521i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.00000i	0.159617i	0.996810	+	0.0798087i	$0.0254309\pi$
$157$	2.00000i	0.159617i	−0.996810	+	0.0798087i	$0.974569\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−16.0000	−1.26098
$162$	0	0
$163$	10.0000i	0.783260i	0.920123	+	0.391630i	$0.128089\pi$
$163$	10.0000i	0.783260i	−0.920123	+	0.391630i	$0.871911\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.00000i	0.464294i	0.972681	+	0.232147i	$0.0745750\pi$
$167$	6.00000i	0.464294i	−0.972681	+	0.232147i	$0.925425\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	18.0000	1.37649
$172$	0	0
$173$	10.0000i	0.760286i	0.924928	+	0.380143i	$0.124125\pi$
$173$	10.0000i	0.760286i	−0.924928	+	0.380143i	$0.875875\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 12.0000i	− 0.877527i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	0	0
$193$	− 2.00000i	− 0.143963i	−0.997406	−	0.0719816i	$-0.977068\pi$
$193$	− 2.00000i	− 0.143963i	0.997406	−	0.0719816i	$-0.0229323\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 6.00000i	− 0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$	− 6.00000i	− 0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	4.00000i	0.280745i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 24.0000i	− 1.66812i
$208$	0	0
$209$	−12.0000	−0.830057
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 20.0000i	− 1.35769i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−6.00000	−0.403604
$222$	0	0
$223$	6.00000i	0.401790i	0.979613	+	0.200895i	$0.0643850\pi$
$223$	6.00000i	0.401790i	−0.979613	+	0.200895i	$0.935615\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	18.0000i	1.19470i	0.801980	+	0.597351i	$0.203780\pi$
$227$	18.0000i	1.19470i	−0.801980	+	0.597351i	$0.796220\pi$
$228$	0	0
$229$	−18.0000	−1.18947	−0.594737	−	0.803921i	$-0.702744\pi$
$229$	−18.0000	−1.18947	−0.594737	+	0.803921i	$0.702744\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 10.0000i	− 0.655122i	−0.944830	−	0.327561i	$-0.893773\pi$
$233$	− 10.0000i	− 0.655122i	0.944830	−	0.327561i	$-0.106227\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−18.0000	−1.16432	−0.582162	−	0.813073i	$-0.697793\pi$
$239$	−18.0000	−1.16432	−0.582162	+	0.813073i	$0.697793\pi$
$240$	0	0
$241$	−6.00000	−0.386494	−0.193247	−	0.981150i	$-0.561902\pi$
$241$	−6.00000	−0.386494	−0.193247	+	0.981150i	$0.561902\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6.00000i	0.381771i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	16.0000i	1.00591i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.0000i	1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$	18.0000i	1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$	0	0
$259$	−12.0000	−0.745644
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	12.0000i	0.739952i	0.929041	+	0.369976i	$0.120634\pi$
$263$	12.0000i	0.739952i	−0.929041	+	0.369976i	$0.879366\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	−2.00000	−0.121491	−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000	−0.121491	−0.0607457	+	0.998153i	$0.519348\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	30.0000i	1.80253i	0.433273	+	0.901263i	$0.357359\pi$
$277$	30.0000i	1.80253i	−0.433273	+	0.901263i	$0.642641\pi$
$278$	0	0
$279$	30.0000	1.79605
$280$	0	0
$281$	−14.0000	−0.835170	−0.417585	−	0.908638i	$-0.637123\pi$
$281$	−14.0000	−0.835170	−0.417585	+	0.908638i	$0.637123\pi$
$282$	0	0
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	12.0000i	0.708338i
$288$	0	0
$289$	−19.0000	−1.11765
$290$	0	0
$291$	0	0
$292$	0	0
$293$	14.0000i	0.817889i	0.912559	+	0.408944i	$0.134103\pi$
$293$	14.0000i	0.817889i	−0.912559	+	0.408944i	$0.865897\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	8.00000	0.462652
$300$	0	0
$301$	−8.00000	−0.461112
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 34.0000i	− 1.94048i	−0.242140	−	0.970241i	$-0.577849\pi$
$307$	− 34.0000i	− 1.94048i	0.242140	−	0.970241i	$-0.422151\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	10.0000i	0.565233i	0.959233	+	0.282617i	$0.0912024\pi$
$313$	10.0000i	0.565233i	−0.959233	+	0.282617i	$0.908798\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	18.0000i	1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$	18.0000i	1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$	0	0
$319$	4.00000	0.223957
$320$	0	0
$321$	0	0
$322$	0	0
$323$	36.0000i	2.00309i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−4.00000	−0.220527
$330$	0	0
$331$	−22.0000	−1.20923	−0.604615	−	0.796518i	$-0.706673\pi$
$331$	−22.0000	−1.20923	−0.604615	+	0.796518i	$0.706673\pi$
$332$	0	0
$333$	− 18.0000i	− 0.986394i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 10.0000i	− 0.544735i	−0.962193	−	0.272367i	$-0.912193\pi$
$337$	− 10.0000i	− 0.544735i	0.962193	−	0.272367i	$-0.0878066\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−20.0000	−1.08306
$342$	0	0
$343$	− 20.0000i	− 1.07990i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 24.0000i	− 1.28839i	−0.764862	−	0.644194i	$-0.777193\pi$
$347$	− 24.0000i	− 1.28839i	0.764862	−	0.644194i	$-0.222807\pi$
$348$	0	0
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	14.0000i	0.745145i	0.928003	+	0.372572i	$0.121524\pi$
$353$	14.0000i	0.745145i	−0.928003	+	0.372572i	$0.878476\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−30.0000	−1.58334	−0.791670	−	0.610949i	$-0.790788\pi$
$359$	−30.0000	−1.58334	−0.791670	+	0.610949i	$0.790788\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 4.00000i	− 0.208798i	−0.994535	−	0.104399i	$-0.966708\pi$
$367$	− 4.00000i	− 0.208798i	0.994535	−	0.104399i	$-0.0332919\pi$
$368$	0	0
$369$	−18.0000	−0.937043
$370$	0	0
$371$	−12.0000	−0.623009
$372$	0	0
$373$	22.0000i	1.13912i	0.821951	+	0.569558i	$0.192886\pi$
$373$	22.0000i	1.13912i	−0.821951	+	0.569558i	$0.807114\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 2.00000i	− 0.103005i
$378$	0	0
$379$	−10.0000	−0.513665	−0.256833	−	0.966456i	$-0.582679\pi$
$379$	−10.0000	−0.513665	−0.256833	+	0.966456i	$0.582679\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 18.0000i	− 0.919757i	−0.887982	−	0.459879i	$-0.847893\pi$
$383$	− 18.0000i	− 0.919757i	0.887982	−	0.459879i	$-0.152107\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 12.0000i	− 0.609994i
$388$	0	0
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	48.0000	2.42746
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2.00000i	0.100377i	0.998740	+	0.0501886i	$0.0159822\pi$
$397$	2.00000i	0.100377i	−0.998740	+	0.0501886i	$0.984018\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	10.0000i	0.498135i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	12.0000i	0.594818i
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 20.0000i	− 0.984136i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−24.0000	−1.17248	−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000	−1.17248	−0.586238	+	0.810139i	$0.699392\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	0	0
$423$	− 6.00000i	− 0.291730i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4.00000i	0.193574i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−30.0000	−1.44505	−0.722525	−	0.691345i	$-0.757018\pi$
$431$	−30.0000	−1.44505	−0.722525	+	0.691345i	$0.757018\pi$
$432$	0	0
$433$	14.0000i	0.672797i	0.941720	+	0.336399i	$0.109209\pi$
$433$	14.0000i	0.672797i	−0.941720	+	0.336399i	$0.890791\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 48.0000i	− 2.29615i
$438$	0	0
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	0	0
$443$	24.0000i	1.14027i	0.821549	+	0.570137i	$0.193110\pi$
$443$	24.0000i	1.14027i	−0.821549	+	0.570137i	$0.806890\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	38.0000	1.79333	0.896665	−	0.442709i	$-0.145982\pi$
$449$	38.0000	1.79333	0.896665	+	0.442709i	$0.145982\pi$
$450$	0	0
$451$	12.0000	0.565058
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	34.0000i	1.59045i	0.606313	+	0.795226i	$0.292648\pi$
$457$	34.0000i	1.59045i	−0.606313	+	0.795226i	$0.707352\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	− 22.0000i	− 1.02243i	−0.859454	−	0.511213i	$-0.829196\pi$
$463$	− 22.0000i	− 1.02243i	0.859454	−	0.511213i	$-0.170804\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	20.0000i	0.925490i	0.886492	+	0.462745i	$0.153135\pi$
$467$	20.0000i	0.925490i	−0.886492	+	0.462745i	$0.846865\pi$
$468$	0	0
$469$	20.0000	0.923514
$470$	0	0
$471$	0	0
$472$	0	0
$473$	8.00000i	0.367840i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 18.0000i	− 0.824163i
$478$	0	0
$479$	18.0000	0.822441	0.411220	−	0.911536i	$-0.365103\pi$
$479$	18.0000	0.822441	0.411220	+	0.911536i	$0.365103\pi$
$480$	0	0
$481$	6.00000	0.273576
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2.00000i	0.0906287i	0.998973	+	0.0453143i	$0.0144289\pi$
$487$	2.00000i	0.0906287i	−0.998973	+	0.0453143i	$0.985571\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	0	0
$493$	− 12.0000i	− 0.540453i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 20.0000i	− 0.897123i
$498$	0	0
$499$	14.0000	0.626726	0.313363	−	0.949633i	$-0.398544\pi$
$499$	14.0000	0.626726	0.313363	+	0.949633i	$0.398544\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 28.0000i	− 1.24846i	−0.781241	−	0.624229i	$-0.785413\pi$
$503$	− 28.0000i	− 1.24846i	0.781241	−	0.624229i	$-0.214587\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	14.0000	0.620539	0.310270	−	0.950649i	$-0.399581\pi$
$509$	14.0000	0.620539	0.310270	+	0.950649i	$0.399581\pi$
$510$	0	0
$511$	−4.00000	−0.176950
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	4.00000i	0.175920i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	− 24.0000i	− 1.04945i	−0.851273	−	0.524723i	$-0.824169\pi$
$523$	− 24.0000i	− 1.04945i	0.851273	−	0.524723i	$-0.175831\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	60.0000i	2.61364i
$528$	0	0
$529$	−41.0000	−1.78261
$530$	0	0
$531$	30.0000	1.30189
$532$	0	0
$533$	− 6.00000i	− 0.259889i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−6.00000	−0.258438
$540$	0	0
$541$	26.0000	1.11783	0.558914	−	0.829226i	$-0.311218\pi$
$541$	26.0000	1.11783	0.558914	+	0.829226i	$0.311218\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 32.0000i	− 1.36822i	−0.729378	−	0.684111i	$-0.760191\pi$
$547$	− 32.0000i	− 1.36822i	0.729378	−	0.684111i	$-0.239809\pi$
$548$	0	0
$549$	−6.00000	−0.256074
$550$	0	0
$551$	−12.0000	−0.511217
$552$	0	0
$553$	− 8.00000i	− 0.340195i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	42.0000i	1.77960i	0.456354	+	0.889799i	$0.349155\pi$
$557$	42.0000i	1.77960i	−0.456354	+	0.889799i	$0.650845\pi$
$558$	0	0
$559$	4.00000	0.169182
$560$	0	0
$561$	0	0
$562$	0	0
$563$	4.00000i	0.168580i	0.996441	+	0.0842900i	$0.0268622\pi$
$563$	4.00000i	0.168580i	−0.996441	+	0.0842900i	$0.973138\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 18.0000i	− 0.755929i
$568$	0	0
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 6.00000i	− 0.249783i	−0.992170	−	0.124892i	$-0.960142\pi$
$577$	− 6.00000i	− 0.249783i	0.992170	−	0.124892i	$-0.0398583\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	12.0000i	0.496989i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	26.0000i	1.07313i	0.843857	+	0.536567i	$0.180279\pi$
$587$	26.0000i	1.07313i	−0.843857	+	0.536567i	$0.819721\pi$
$588$	0	0
$589$	60.0000	2.47226
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 26.0000i	− 1.06769i	−0.845582	−	0.533846i	$-0.820746\pi$
$593$	− 26.0000i	− 1.06769i	0.845582	−	0.533846i	$-0.179254\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−4.00000	−0.163436	−0.0817178	−	0.996656i	$-0.526041\pi$
$599$	−4.00000	−0.163436	−0.0817178	+	0.996656i	$0.526041\pi$
$600$	0	0
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	0	0
$603$	30.0000i	1.22169i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 12.0000i	− 0.487065i	−0.969893	−	0.243532i	$-0.921694\pi$
$607$	− 12.0000i	− 0.487065i	0.969893	−	0.243532i	$-0.0783062\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.00000	0.0809113
$612$	0	0
$613$	− 26.0000i	− 1.05013i	−0.851062	−	0.525065i	$-0.824041\pi$
$613$	− 26.0000i	− 1.05013i	0.851062	−	0.525065i	$-0.175959\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	34.0000i	1.36879i	0.729112	+	0.684394i	$0.239933\pi$
$617$	34.0000i	1.36879i	−0.729112	+	0.684394i	$0.760067\pi$
$618$	0	0
$619$	2.00000	0.0803868	0.0401934	−	0.999192i	$-0.487203\pi$
$619$	2.00000	0.0803868	0.0401934	+	0.999192i	$0.487203\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 12.0000i	− 0.480770i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	36.0000	1.43541
$630$	0	0
$631$	22.0000	0.875806	0.437903	−	0.899022i	$-0.355721\pi$
$631$	22.0000	0.875806	0.437903	+	0.899022i	$0.355721\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3.00000i	0.118864i
$638$	0	0
$639$	30.0000	1.18678
$640$	0	0
$641$	−26.0000	−1.02694	−0.513469	−	0.858108i	$-0.671640\pi$
$641$	−26.0000	−1.02694	−0.513469	+	0.858108i	$0.671640\pi$
$642$	0	0
$643$	− 2.00000i	− 0.0788723i	−0.999222	−	0.0394362i	$-0.987444\pi$
$643$	− 2.00000i	− 0.0788723i	0.999222	−	0.0394362i	$-0.0125562\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 32.0000i	− 1.25805i	−0.777385	−	0.629025i	$-0.783454\pi$
$647$	− 32.0000i	− 1.25805i	0.777385	−	0.629025i	$-0.216546\pi$
$648$	0	0
$649$	−20.0000	−0.785069
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 18.0000i	− 0.704394i	−0.935926	−	0.352197i	$-0.885435\pi$
$653$	− 18.0000i	− 0.704394i	0.935926	−	0.352197i	$-0.114565\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 6.00000i	− 0.234082i
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	2.00000	0.0777910	0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000	0.0777910	0.0388955	+	0.999243i	$0.487616\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	16.0000i	0.619522i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	4.00000	0.154418
$672$	0	0
$673$	26.0000i	1.00223i	0.865382	+	0.501113i	$0.167076\pi$
$673$	26.0000i	1.00223i	−0.865382	+	0.501113i	$0.832924\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6.00000i	0.230599i	0.993331	+	0.115299i	$0.0367827\pi$
$677$	6.00000i	0.230599i	−0.993331	+	0.115299i	$0.963217\pi$
$678$	0	0
$679$	4.00000	0.153506
$680$	0	0
$681$	0	0
$682$	0	0
$683$	2.00000i	0.0765279i	0.999268	+	0.0382639i	$0.0121828\pi$
$683$	2.00000i	0.0765279i	−0.999268	+	0.0382639i	$0.987817\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	6.00000	0.228582
$690$	0	0
$691$	34.0000	1.29342	0.646710	−	0.762736i	$-0.276144\pi$
$691$	34.0000	1.29342	0.646710	+	0.762736i	$0.276144\pi$
$692$	0	0
$693$	12.0000i	0.455842i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 36.0000i	− 1.36360i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	− 36.0000i	− 1.35777i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.00000i	0.150435i
$708$	0	0
$709$	22.0000	0.826227	0.413114	−	0.910679i	$-0.364441\pi$
$709$	22.0000	0.826227	0.413114	+	0.910679i	$0.364441\pi$
$710$	0	0
$711$	12.0000	0.450035
$712$	0	0
$713$	− 80.0000i	− 2.99602i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	40.0000	1.49175	0.745874	−	0.666087i	$-0.232032\pi$
$719$	40.0000	1.49175	0.745874	+	0.666087i	$0.232032\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 32.0000i	− 1.18681i	−0.804902	−	0.593407i	$-0.797782\pi$
$727$	− 32.0000i	− 1.18681i	0.804902	−	0.593407i	$-0.202218\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	24.0000	0.887672
$732$	0	0
$733$	− 26.0000i	− 0.960332i	−0.877178	−	0.480166i	$-0.840576\pi$
$733$	− 26.0000i	− 0.960332i	0.877178	−	0.480166i	$-0.159424\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 20.0000i	− 0.736709i
$738$	0	0
$739$	−6.00000	−0.220714	−0.110357	−	0.993892i	$-0.535199\pi$
$739$	−6.00000	−0.220714	−0.110357	+	0.993892i	$0.535199\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	14.0000i	0.513610i	0.966463	+	0.256805i	$0.0826698\pi$
$743$	14.0000i	0.513610i	−0.966463	+	0.256805i	$0.917330\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	18.0000i	0.658586i
$748$	0	0
$749$	−32.0000	−1.16925
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 42.0000i	− 1.52652i	−0.646094	−	0.763258i	$-0.723599\pi$
$757$	− 42.0000i	− 1.52652i	0.646094	−	0.763258i	$-0.276401\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	0	0
$763$	− 28.0000i	− 1.01367i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	10.0000i	0.361079i
$768$	0	0
$769$	54.0000	1.94729	0.973645	−	0.228069i	$-0.0732413\pi$
$769$	54.0000	1.94729	0.973645	+	0.228069i	$0.0732413\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	46.0000i	1.65451i	0.561830	+	0.827253i	$0.310097\pi$
$773$	46.0000i	1.65451i	−0.561830	+	0.827253i	$0.689903\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−36.0000	−1.28983
$780$	0	0
$781$	−20.0000	−0.715656
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	38.0000i	1.35455i	0.735728	+	0.677277i	$0.236840\pi$
$787$	38.0000i	1.35455i	−0.735728	+	0.677277i	$0.763160\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−28.0000	−0.995565
$792$	0	0
$793$	− 2.00000i	− 0.0710221i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	18.0000i	0.637593i	0.947823	+	0.318796i	$0.103279\pi$
$797$	18.0000i	0.637593i	−0.947823	+	0.318796i	$0.896721\pi$
$798$	0	0
$799$	12.0000	0.424529
$800$	0	0
$801$	18.0000	0.635999
$802$	0	0
$803$	4.00000i	0.141157i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	−22.0000	−0.772524	−0.386262	−	0.922389i	$-0.626234\pi$
$811$	−22.0000	−0.772524	−0.386262	+	0.922389i	$0.626234\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 24.0000i	− 0.839654i
$818$	0	0
$819$	6.00000	0.209657
$820$	0	0
$821$	2.00000	0.0698005	0.0349002	−	0.999391i	$-0.488889\pi$
$821$	2.00000	0.0698005	0.0349002	+	0.999391i	$0.488889\pi$
$822$	0	0
$823$	24.0000i	0.836587i	0.908312	+	0.418294i	$0.137372\pi$
$823$	24.0000i	0.836587i	−0.908312	+	0.418294i	$0.862628\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	6.00000i	0.208640i	0.994544	+	0.104320i	$0.0332667\pi$
$827$	6.00000i	0.208640i	−0.994544	+	0.104320i	$0.966733\pi$
$828$	0	0
$829$	−22.0000	−0.764092	−0.382046	−	0.924143i	$-0.624780\pi$
$829$	−22.0000	−0.764092	−0.382046	+	0.924143i	$0.624780\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	18.0000i	0.623663i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	10.0000	0.345238	0.172619	−	0.984989i	$-0.444777\pi$
$839$	10.0000	0.345238	0.172619	+	0.984989i	$0.444777\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	14.0000i	0.481046i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−48.0000	−1.64542
$852$	0	0
$853$	38.0000i	1.30110i	0.759465	+	0.650548i	$0.225461\pi$
$853$	38.0000i	1.30110i	−0.759465	+	0.650548i	$0.774539\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 38.0000i	− 1.29806i	−0.760765	−	0.649028i	$-0.775176\pi$
$857$	− 38.0000i	− 1.29806i	0.760765	−	0.649028i	$-0.224824\pi$
$858$	0	0
$859$	−8.00000	−0.272956	−0.136478	−	0.990643i	$-0.543578\pi$
$859$	−8.00000	−0.272956	−0.136478	+	0.990643i	$0.543578\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 34.0000i	− 1.15737i	−0.815550	−	0.578687i	$-0.803565\pi$
$863$	− 34.0000i	− 1.15737i	0.815550	−	0.578687i	$-0.196435\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−8.00000	−0.271381
$870$	0	0
$871$	−10.0000	−0.338837
$872$	0	0
$873$	6.00000i	0.203069i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	18.0000i	0.607817i	0.952701	+	0.303908i	$0.0982917\pi$
$877$	18.0000i	0.607817i	−0.952701	+	0.303908i	$0.901708\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	− 28.0000i	− 0.942275i	−0.882060	−	0.471138i	$-0.843844\pi$
$883$	− 28.0000i	− 0.942275i	0.882060	−	0.471138i	$-0.156156\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	4.00000i	0.134307i	0.997743	+	0.0671534i	$0.0213917\pi$
$887$	4.00000i	0.134307i	−0.997743	+	0.0671534i	$0.978608\pi$
$888$	0	0
$889$	−16.0000	−0.536623
$890$	0	0
$891$	−18.0000	−0.603023
$892$	0	0
$893$	− 12.0000i	− 0.401565i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−20.0000	−0.667037
$900$	0	0
$901$	36.0000	1.19933
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	36.0000i	1.19536i	0.801735	+	0.597680i	$0.203911\pi$
$907$	36.0000i	1.19536i	−0.801735	+	0.597680i	$0.796089\pi$
$908$	0	0
$909$	−6.00000	−0.199007
$910$	0	0
$911$	−36.0000	−1.19273	−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000	−1.19273	−0.596367	+	0.802712i	$0.703390\pi$
$912$	0	0
$913$	− 12.0000i	− 0.397142i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	32.0000i	1.05673i
$918$	0	0
$919$	12.0000	0.395843	0.197922	−	0.980218i	$-0.436581\pi$
$919$	12.0000	0.395843	0.197922	+	0.980218i	$0.436581\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	10.0000i	0.329154i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	24.0000i	0.788263i
$928$	0	0
$929$	54.0000	1.77168	0.885841	−	0.463988i	$-0.153582\pi$
$929$	54.0000	1.77168	0.885841	+	0.463988i	$0.153582\pi$
$930$	0	0
$931$	18.0000	0.589926
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	42.0000i	1.37208i	0.727564	+	0.686040i	$0.240653\pi$
$937$	42.0000i	1.37208i	−0.727564	+	0.686040i	$0.759347\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	10.0000	0.325991	0.162995	−	0.986627i	$-0.447884\pi$
$941$	10.0000	0.325991	0.162995	+	0.986627i	$0.447884\pi$
$942$	0	0
$943$	48.0000i	1.56310i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 42.0000i	− 1.36482i	−0.730971	−	0.682408i	$-0.760933\pi$
$947$	− 42.0000i	− 1.36482i	0.730971	−	0.682408i	$-0.239067\pi$
$948$	0	0
$949$	2.00000	0.0649227
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 26.0000i	− 0.842223i	−0.907009	−	0.421111i	$-0.861640\pi$
$953$	− 26.0000i	− 0.842223i	0.907009	−	0.421111i	$-0.138360\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	36.0000	1.16250
$960$	0	0
$961$	69.0000	2.22581
$962$	0	0
$963$	− 48.0000i	− 1.54678i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	50.0000i	1.60789i	0.594703	+	0.803946i	$0.297270\pi$
$967$	50.0000i	1.60789i	−0.594703	+	0.803946i	$0.702730\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−48.0000	−1.54039	−0.770197	−	0.637806i	$-0.779842\pi$
$971$	−48.0000	−1.54039	−0.770197	+	0.637806i	$0.779842\pi$
$972$	0	0
$973$	32.0000i	1.02587i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	18.0000i	0.575871i	0.957650	+	0.287936i	$0.0929689\pi$
$977$	18.0000i	0.575871i	−0.957650	+	0.287936i	$0.907031\pi$
$978$	0	0
$979$	−12.0000	−0.383522
$980$	0	0
$981$	42.0000	1.34096
$982$	0	0
$983$	− 14.0000i	− 0.446531i	−0.974758	−	0.223265i	$-0.928328\pi$
$983$	− 14.0000i	− 0.446531i	0.974758	−	0.223265i	$-0.0716716\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−32.0000	−1.01754
$990$	0	0
$991$	12.0000	0.381193	0.190596	−	0.981669i	$-0.438958\pi$
$991$	12.0000	0.381193	0.190596	+	0.981669i	$0.438958\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 58.0000i	− 1.83688i	−0.395562	−	0.918439i	$-0.629450\pi$
$997$	− 58.0000i	− 1.83688i	0.395562	−	0.918439i	$-0.370550\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1300.2.c.c.1249.1		2
5.2	odd	4		1300.2.a.d.1.1		1
5.3	odd	4		52.2.a.a.1.1	✓	1
5.4	even	2	inner	1300.2.c.c.1249.2		2
15.8	even	4		468.2.a.b.1.1		1
20.3	even	4		208.2.a.c.1.1		1
20.7	even	4		5200.2.a.q.1.1		1
35.3	even	12		2548.2.j.f.1353.1		2
35.13	even	4		2548.2.a.e.1.1		1
35.18	odd	12		2548.2.j.e.1353.1		2
35.23	odd	12		2548.2.j.e.1145.1		2
35.33	even	12		2548.2.j.f.1145.1		2
40.3	even	4		832.2.a.f.1.1		1
40.13	odd	4		832.2.a.e.1.1		1
45.13	odd	12		4212.2.i.d.2809.1		2
45.23	even	12		4212.2.i.i.2809.1		2
45.38	even	12		4212.2.i.i.1405.1		2
45.43	odd	12		4212.2.i.d.1405.1		2
55.43	even	4		6292.2.a.g.1.1		1
60.23	odd	4		1872.2.a.f.1.1		1
65.3	odd	12		676.2.e.c.529.1		2
65.8	even	4		676.2.d.c.337.2		2
65.18	even	4		676.2.d.c.337.1		2
65.23	odd	12		676.2.e.b.529.1		2
65.28	even	12		676.2.h.c.485.1		4
65.33	even	12		676.2.h.c.361.2		4
65.38	odd	4		676.2.a.c.1.1		1
65.43	odd	12		676.2.e.b.653.1		2
65.48	odd	12		676.2.e.c.653.1		2
65.58	even	12		676.2.h.c.361.1		4
65.63	even	12		676.2.h.c.485.2		4
80.3	even	4		3328.2.b.e.1665.1		2
80.13	odd	4		3328.2.b.q.1665.1		2
80.43	even	4		3328.2.b.e.1665.2		2
80.53	odd	4		3328.2.b.q.1665.2		2
120.53	even	4		7488.2.a.bn.1.1		1
120.83	odd	4		7488.2.a.bw.1.1		1
195.8	odd	4		6084.2.b.m.4393.1		2
195.38	even	4		6084.2.a.m.1.1		1
195.83	odd	4		6084.2.b.m.4393.2		2
260.83	odd	4		2704.2.f.f.337.1		2
260.103	even	4		2704.2.a.g.1.1		1
260.203	odd	4		2704.2.f.f.337.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
52.2.a.a.1.1	✓	1	5.3	odd	4
208.2.a.c.1.1		1	20.3	even	4
468.2.a.b.1.1		1	15.8	even	4
676.2.a.c.1.1		1	65.38	odd	4
676.2.d.c.337.1		2	65.18	even	4
676.2.d.c.337.2		2	65.8	even	4
676.2.e.b.529.1		2	65.23	odd	12
676.2.e.b.653.1		2	65.43	odd	12
676.2.e.c.529.1		2	65.3	odd	12
676.2.e.c.653.1		2	65.48	odd	12
676.2.h.c.361.1		4	65.58	even	12
676.2.h.c.361.2		4	65.33	even	12
676.2.h.c.485.1		4	65.28	even	12
676.2.h.c.485.2		4	65.63	even	12
832.2.a.e.1.1		1	40.13	odd	4
832.2.a.f.1.1		1	40.3	even	4
1300.2.a.d.1.1		1	5.2	odd	4
1300.2.c.c.1249.1		2	1.1	even	1	trivial
1300.2.c.c.1249.2		2	5.4	even	2	inner
1872.2.a.f.1.1		1	60.23	odd	4
2548.2.a.e.1.1		1	35.13	even	4
2548.2.j.e.1145.1		2	35.23	odd	12
2548.2.j.e.1353.1		2	35.18	odd	12
2548.2.j.f.1145.1		2	35.33	even	12
2548.2.j.f.1353.1		2	35.3	even	12
2704.2.a.g.1.1		1	260.103	even	4
2704.2.f.f.337.1		2	260.83	odd	4
2704.2.f.f.337.2		2	260.203	odd	4
3328.2.b.e.1665.1		2	80.3	even	4
3328.2.b.e.1665.2		2	80.43	even	4
3328.2.b.q.1665.1		2	80.13	odd	4
3328.2.b.q.1665.2		2	80.53	odd	4
4212.2.i.d.1405.1		2	45.43	odd	12
4212.2.i.d.2809.1		2	45.13	odd	12
4212.2.i.i.1405.1		2	45.38	even	12
4212.2.i.i.2809.1		2	45.23	even	12
5200.2.a.q.1.1		1	20.7	even	4
6084.2.a.m.1.1		1	195.38	even	4
6084.2.b.m.4393.1		2	195.8	odd	4
6084.2.b.m.4393.2		2	195.83	odd	4
6292.2.a.g.1.1		1	55.43	even	4
7488.2.a.bn.1.1		1	120.53	even	4
7488.2.a.bw.1.1		1	120.83	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1300.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q-2.00000i q^{7} +3.00000 q^{9} -2.00000 q^{11} +1.00000i q^{13} +6.00000i q^{17} +6.00000 q^{19} -8.00000i q^{23} -2.00000 q^{29} +10.0000 q^{31} -6.00000i q^{37} -6.00000 q^{41} -4.00000i q^{43} -2.00000i q^{47} +3.00000 q^{49} -6.00000i q^{53} +10.0000 q^{59} -2.00000 q^{61} -6.00000i q^{63} +10.0000i q^{67} +10.0000 q^{71} -2.00000i q^{73} +4.00000i q^{77} +4.00000 q^{79} +9.00000 q^{81} +6.00000i q^{83} +6.00000 q^{89} +2.00000 q^{91} +2.00000i q^{97} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{9}+O(q^{10}) \) 2 * q + 6 * q^9	\( 2 q + 6 q^{9} - 4 q^{11} + 12 q^{19} - 4 q^{29} + 20 q^{31} - 12 q^{41} + 6 q^{49} + 20 q^{59} - 4 q^{61} + 20 q^{71} + 8 q^{79} + 18 q^{81} + 12 q^{89} + 4 q^{91} - 12 q^{99}+O(q^{100}) \) 2 * q + 6 * q^9 - 4 * q^11 + 12 * q^19 - 4 * q^29 + 20 * q^31 - 12 * q^41 + 6 * q^49 + 20 * q^59 - 4 * q^61 + 20 * q^71 + 8 * q^79 + 18 * q^81 + 12 * q^89 + 4 * q^91 - 12 * q^99

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